Scientific Papers - Vi

484
ON PERIODIC IBROTATIONAL WAVES
[419
In the fourth column are recorded the values of Z72 —2#y —1, calculated from (1) with omission of 8 and e, and with the corrected values of y. dty/dx, ffy//<% were first found separately, and then U2 as the sum of the two squares. The values of /3, 7, g employed are those given in (15), (18), (20). The form of -Jr in (1) with these values of the constants vanishes when y takes the values of the third column, and the pressure at the surface is also constant to a high degree of approximation. The greatest difference is ('000,001,0), which may be compared with '4, the latter amount representing the corresponding statical difference at the crest and trough of the wave. According to this standard the pressure at the surface is constant to 2£ parts in a million*.
The advantage gained by the introduction of /3 and y will be better estimated by comparison with a similar calculation where only a (still equal to T17) and g are retained. By (2) in this case
U*-2gy-I**e?r*» + 2(l-g)y...................(23)
: Table II shows the values of y and of «V~22/ corresponding to the same values of as as before. The fourth column gives (23) when g is so determined as to make the values equal at 0° and 180°. It appears that the discrepancy in the values of U2 — 2gy is reduced 200 times by the introduction of /3 and 7, even when we tie ourselves to the values of /3, 7, g prescribed by approximations on the lines of Stokes.
TABLE II.
X y a2e-2z/ D2 -2gy-l
0 + •091,276,5 •008,331,4 •010,207,7
22* •084,870,5 •008,438,8 . 183,4
45" •066,182,4 •008,760,2 . 120,7
67* •036,882,6 •009,288,9 . 047,1
90" 0 •010,000,0 . 000,0
112* - -039,823,1 •010,829,0 . 010,4
135" - -076,318,5 •011,649,0 . 080,2
157* -•102,344,1 •012,271,4 . 167,6
180" -•111,832,6 •012,506,5 •010,207,7
A cursory inspection of the numbers in column 4 of Table I suffices to show that an improvement can be effected by a slight alteration in the value of /3. For small corrections of this kind it is convenient to use a formula which may be derived from (2). We suppose that while a and ty are maintained constant, small alterations 8/3, 87, Sg are incurred. Neglecting the small variations of ft, 7, g when multiplied by a2 and higher powers of a, we get
By = 8/3 jcos 2#'— &a cos x — £a cos 3#}
t/ ' t, fi & )
+ 87 {cos 3& - 2a cos 2x — 2a cos 4*},...............(24)
[* "With the alterations specified in footnote J on p. 483, the greatest difference becomes •000,001,1, so that the surface pressure is constant to 2J parts in a million. W. F. S.] network of conductors, it gt-nerates less current, and accordingly does less work, when an interruption occurs, as by breaking a contact in any part of the system.f gelatine, scarcely thicker than thick paper, should be able to tear out fragments of-solid glass, but there is no doubt of the fact.